On a new digraph reconstruction conjecture

Some classes of digraphs are reconstructed from the point-deleted subdigraphs for each of which the degree pair of the deleted point is also known. Several infinite families of known counterexamples to the Digraph Reconstruction Conjecture (DRC) turn out to be reconstructible in this sense. A new conjecture concerning reconstruction of digraphs in this sense is proposed and none of the known counterexample pairs to the DRC is a counterexample pair to this new conjecture

An algebraic formulation of the graph reconstruction conjecture

The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph $G$ and any finite sequence of graphs, it gives a linear constraint that every reconstruction of $G$ must satisfy. Let $\psi(n)$ be the number of distinct (mutually non-isomorphic) graphs on $n$ vertices, and let $d(n)$ be the number of distinct decks that can be constructed from these graphs. Then the difference $\psi(n) - d(n)$ measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for $n$-vertex graphs if and only if $\psi(n) = d(n)$. We give a framework based on Kocay's lemma to study this discrepancy. We prove that if $M$ is a matrix of covering numbers of graphs by sequences of graphs, then $d(n) \geq \mathsf{rank}_\mathbb{R}(M)$. In particular, all $n$-vertex graphs are reconstructible if one such matrix has rank $\psi(n)$. To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix $M$ of covering numbers satisfies $d(n) = \mathsf{rank}_\mathbb{R}(M)$.Comment: 12 pages, 2 figure

Hypomorphisms, orbits, and reconstruction

AbstractGraphs G and H are hypomorphic if there is a bijection φ: V(G) → V(H) such that G − u ≅ H − φ(u), for all u ∈ V(G). The reconstruction conjecture states that hypomorphic graphs are isomorphic, if G has at least three vertices. We investigate properties of the isomorphisms G − u ≅ H − φ(u), and their relation to the reconstructibility of G

On the Reconstruction Conjecture

"Every graph of order three or more is reconstructible." Frank Harary restated one of the most famous unsolved problems in graph theory. In the early 1900's, while one was working on his doctoral dissertation, two mathematicians made a conjecture about the reconstructibility of graphs. This came to be known as the Reconstruction Conjecture or the Kelly-Ulam Conjecture. The conjecture states: Let G and H be graphs with V(G) = {v_1, v_2, ..., v_n}, V(H) = {u_1, u_2, ..., u_n}, n greater than or equal to 3. If G - v_i is isomorphic to H - u_i for all i = 1, ..., n, then G is isomorphic to H. Much progress has been made toward showing that this statement is true for all graphs. This paper will discuss some of that progress, including some of the families of graphs which we know that the conjecture is true. Another big field of interest about the Reconstruction Conjecture is the information that is retained by a graph when we begin looking at its vertex-deleted subgraphs. Many graph theorists believe that this may show us more about the conjecture as a whole. While working on a possible proof to the Reconstruction Conjecture, many mathematicians began to think about different approaches. One approach that was fairly common was to relate the Reconstruction Conjecture to edges of a graph instead of the vertices. People realized that when deleting only one edge of a graph, then logically more information about the original graph would be retained